RD Sharma Solutions for Class 12, maths Chapter 19

Class 12: Maths Chapter 19 solutions. Complete Class 12 Maths Chapter 19 Notes.

Contents

1 RD Sharma Solutions for Class 12 Maths Chapter 19–Indefinite Integrals
2 RD Sharma Solutions for Class 12 Maths Chapter 19: Download PDF
3 Chapterwise RD Sharma Solutions for Class 12 Maths :
4 About RD Sharma

RD Sharma Solutions for Class 12 Maths Chapter 19–Indefinite Integrals

RD Sharma 12th Maths Chapter 19, Class 12 Maths Chapter 19 solutions

Exercise 19.1 Page No: 19.4

1. Evaluate the following integrals:

Solution:

Given

Solution:

Given

Solution:

Given

Solution:

Given

Solution:

Given

Solution:

Given

Solution:

Given

Solution:

Given

2. Evaluate:

Solution:

Given

Solution:

Given

3. Evaluate:

Solution:

Given

Exercise 19.2 Page No: 19.14

Evaluate the following integrals (1 – 44):

Solution:

Given

Solution:

Given

Solution:

Given,

∫(2 – 3x)(3 + 2x)(1 – 2x) dx

= ∫(6 + 4x – 9x – 6x²)(1 – 2x) dx

= ∫(6 – 5x – 6x²)(1 – 2x) dx

= ∫(6 – 5x – 6x² – 12x + 10x² + 12x³) dx

= ∫(6 – 17x + 4x² + 12x³) dx

Upon splitting the above, we have

= ∫6 dx – ∫17x dx + ∫4x² dx + ∫12x³ dx

On integrating using formula,

∫xⁿ dx = xⁿ⁺¹/n+1

we get

= 6x – 17/(1+1) x¹⁺¹ + 4/(2+1) x²⁺¹ + 12/(3+1) x³⁺¹ + c

= 6x – 17x²/2 + 4x³/3 + 3x⁴ + c

Solution:

Given

Solution:

Given

Solution:

Given

Solution:

Given

Solution:

Given

Solution:

Given

Solution:

Given

Solution:

Given

Solution:

Given

Exercise 19.3 Page No: 19.23

Solution:

Solution:

Exercise 19.4 Page No: 19.30

Solution:

Exercise 19.5 Page No: 19.33

Solution:

Given

Solution:

Exercise 19.6 Page No: 19.36

Solution:

Exercise 19.7 Page No: 19.38

Integrate the following integrals:

Solution:

Exercise 19.8 Page No: 19.47

Evaluate the following integrals:

Solution:

Given,

Solution:

Given,

Solution:

Therefore,

= cos (b – a)x + sin(b – a) log |sin(x – b)| + c, where c is an arbitrary constant.

Exercise 19.9 Page No: 19.57

Evaluate the following integrals:

Solution:

Exercise 19.10 Page No: 19.65

Solution:

Exercise 19.11 Page No: 19.69

Evaluate the following integrals:

Solution:

Exercise 19.12 Page No: 19.73

Solution:

Exercise 19.13 Page No: 19.79

Solution:

Exercise 19.14 Page No: 19.83

Evaluate the following integrals:

Solution:

Exercise 19.15 Page No: 19.86

Solution:

By using,

Solution:

Exercise 19.16 Page No: 19.90

Evaluate the following integrals:

Solution:

Exercise 19.17 Page No: 19.93

Evaluate the following integrals:

Solution:

Exercise 19.18 Page No: 19.98

Evaluate the following integrals:

Solution:

Let sin x = t

Solution:

Exercise 19.19 Page No: 19.104

Evaluate the following integrals:

Solution:

We will solve I₁ and I₂ individually.

Solution:

Exercise 19.20 Page No: 19.106

Evaluate the following integrals:

Solution:

⇒ 1 = (A + B) x + (3A – 2B)

⇒ Then A + B = 0 … (1)

And 3A – 2B = 1 … (2)

Solving (1) and (2),

2 × (1) → 2A + 2B = 0

1 × (2) → 3A – 2B = 1

5A = 1

∴ A = 1/5

Substituting A value in (1),

Or I = log|(x – 2)/(x + 3)| + x + c

Solution:

Hence,

Exercise 19.21 Page No: 19.110

Evaluate the following integrals:

Solution:

Exercise 19.22 Page No: 19.114

Evaluate the following integrals:

Solution:

Exercise 19.23 Page No: 19.117

Evaluate the following integrals:

Solution:

Exercise 19.24 Page No: 19.122

Evaluate the following integrals:

Solution:

Exercise 19.25 Page No: 19.133

Evaluate the following integrals:

Solution:

Solution:

Exercise 19.26 Page No: 19.143

Evaluate the following integrals:

Solution:

Exercise 19.27 Page No: 19.149

Evaluate the following integrals:

Solution:

Exercise 19.28 Page No: 19.154

Evaluate the following integrals:

Solution:

Exercise 19.29 Page No: 19.158

Evaluate the following integrals:

Solution:

Exercise 19.30 Page No: 19.176

Evaluate the following integrals:

Solution:

Exercise 19.31 Page No: 19.190

Evaluate the following integrals:

Solution:

The given equation can be written as,

Solution:

Now, substituting t as x – 1/x and z as x + 1/x we have

Solution:

We get,

Solution:

Exercise 19.32 Page No: 19.196

Evaluate the following integrals:

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 19: Download PDF

RD Sharma Solutions for Class 12 Maths Chapter 19–Indefinite Integrals

Download PDF: RD Sharma Solutions for Class 12 Maths Chapter 19–Indefinite Integrals PDF

Chapterwise RD Sharma Solutions for Class 12 Maths :

About RD Sharma

RD Sharma isn’t the kind of author you’d bump into at lit fests. But his bestselling books have helped many CBSE students lose their dread of maths. Sunday Times profiles the tutor turned internet star
He dreams of algorithms that would give most people nightmares. And, spends every waking hour thinking of ways to explain concepts like ‘series solution of linear differential equations’. Meet Dr Ravi Dutt Sharma — mathematics teacher and author of 25 reference books — whose name evokes as much awe as the subject he teaches. And though students have used his thick tomes for the last 31 years to ace the dreaded maths exam, it’s only recently that a spoof video turned the tutor into a YouTube star.

R D Sharma had a good laugh but said he shared little with his on-screen persona except for the love for maths. “I like to spend all my time thinking and writing about maths problems. I find it relaxing,” he says. When he is not writing books explaining mathematical concepts for classes 6 to 12 and engineering students, Sharma is busy dispensing his duty as vice-principal and head of department of science and humanities at Delhi government’s Guru Nanak Dev Institute of Technology.

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